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VOL. 6, NO. 1, JAN.-MARCH 1992 J. THERMOPHYSICS 121 Numerical Study of Natural Convection from Discrete Heat Sources in a Vertical Square Enclosure G. Refai Ahmed* and M. M. Yovanovichl. University of

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VOL. 6, NO. 1, JAN.-MARCH 1992 J. THERMOPHYSICS 121 Numerical Study of Natural Convection from Discrete Heat Sources in a Vertical Square Enclosure G. Refai Ahmed* and M. M. Yovanovichl. University of Waterloo, Waterloo, Ontario, N2L 3Gl Canada A numerical finite difference technique based on the Marker and Cell (MAC) method is used to obtain solutions of a two-dimensional model of a square enclosure with laminar natural convection heat transfer from discrete heat sources. A discrete heat source is located in the center of one vertical side representing a highpower integrated circuit (IC). The conservation equations are solved using the primitive variables: velocity, pressure, and temperature. Computations are carried out for Pr = 0.72, A = 1 and 0 5 Ra 5 lo6 (Rayleigh number is based on the length of the heat source S divided by the aspect ratio A). The ratio E of the heat source size to the total height lies in the range E Verification of numerical results are obtained at Ra = 0 (conduction limit) with an analytical conduction solution, and the dependence of Nu and total resistance on Ra, E, and boundary conditions are studied. Relationships between Nu and Ra based on different scale lengths are examined. In addition, a relationship between Nu and Ra, based on SIA, are correlated as Nu = Nu (Ra, E) and extrapolation equations are developed to cover the range of Ra from 0 5 Ra lo9. A CP % Dff. = Gr Gr* g H h h k L Nu pd Pr Pd Q 9 Ra Ra* Rc Rm RI ; T t U U Nomenclature = aspect ratio of cavity, H/L = specific heat at constant pressure, kj/kg. K [(RI ,,- RIa a,)/RIa a,l x 100% Grashof number, S3Pg(Th - Tc)/3A3 isoflux Grashof number, S4Pgq/3kA4 gravitational acceleration, m/sz height of the cavity, m coefficient of heat transfer, W/m2. K local coefficient of heat transfer, W/m2. K thermal conductivity, W/m. K width of cavity, m average Nusselt number, hs/ka nondimensional dynamic pressure Prandtl number, v/a dimensional dynamic pressure, N/m2 total heat flow rate, W heat flux at discrete heat source, W/m2 Rayleigh number, S3pg(Th - TJavA3 isoflux Rayleigh number, S4pgq/a~lkA4 nondimensional constriction resistance nondimensional material resistance nondimensional total resistance dimensional total resistance, WW length of discrete heat source, m temperature, K time, s nondimensional velocity component in X direction dimensional velocity component in x direction, m/s Presented as Paper at the AIAA 28th Aerospace Sciences Meeting, Reno, NV, Jan. 8-11, 1990; received March 5, 1990; revision received Sept. 20,1990; accepted for publication Sept. 21,1990. Copyright by G. Refai Ahmed and M. M. Yovanovich. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. *Graduate Research Assistant, Microelectronics Heat Transfer Laboratory, Department of Mechanical Engineering. Member AIAA. tprofessor of Mechanical Engineering and Electrical Engineering, Microelectronics Heat Transfer Laboratory, Department of Mechanical Engineering. Associate Fellow AIAA. = nondimensional velocity component in Y direction = dimensional velocity component in y direction, m/s = nondimensional coordinate = dimensional coordinate, m = nondimensional coordinate = nondimensional length, [A(H- S)/2S] = nondimensional length, [A(H + S)/2S] = nondimensional length, [HAIS] = dimensional coordinate, m = thermal diffusivity, k/c,p, mz/s = coefficient of thermal expansion, 1/K = relative discrete heat source size, S/H = eigenvalues defined in Eq. (6) = kinematic viscosity, mz/s = density, kg/m3 = nondimensional time = nondimensional temperature, T - TJT, - or (T- T,)kA/qS = temperature excess, T - T,, K = area-average source temperature excess, -P rp l,, Subscripts anal = analytical solution C = cold temperature H = height of the cavity h = hot temperature L = width of the cavity num = numerical solution S = discrete heat source Abbreviations GE = governing equation IFDHS = isoflux discrete heat source ITDHS = isothermal discrete heat source MAC = Marker and Cell Mathematical Expressions - D =-+u-i-v- a a a Dt at ax ay 122 G. REFAI AHMED AND M. M. YOVANOVICH J. THERMOPHYSICS Introduction VER the past twenty years, a revolution in electronics 0 has taken place. The miniaturization which resulted from Large Scale Integration (LSI) of components has led to the microminiaturization of Very Large Scale Integration (VLSI). As a consequence of packing a very large number of components into one very small chip, the attendant volumetric heat generation rate has risen to extremely high levels. The power per unit volume that must be dissipated by modern electronic devices is of the same order of magnitude as that of a pressurized water nuclear reactor as noted by Kelleher. Natural convection cooling of components attached to printed circuit boards which are placed vertically in an enclosure is currently of great interest to the microelectronics industry. Natural convection cooling is desirable because it does not require an energy source, such as a forced air fan, and it is maintenance free and safe. The enclosure, as shown in Fig. 1, consists of two vertical boundaries of height H, and two horizontal boundaries of length L. One vertical boundary is cooled at T, and the other has a discrete heat source (isoflux q or isothermal Th) on an otherwise adiabatic surface. The top and bottom horizontal boundaries are adiabatic. The work of Chu and Churchill represents a first contribution to the study of natural convection in an enclosure with concentrated heat sources. In this study a finite difference formulation using a two-dimensional mesh (10 x 10) was employed to solve the transient equations in order to obtain the steady state solution. The solutions presented in Reference 2 are for aspect ratios, A = HIL of 0.4 to 5, Grashof number (based on the height of the cavity) from 0 to lo5, and a Prandtl number of air. The size and location of the heater strip were also varied. More recently Turner and Flack3 and Flack and Turner 4 conducted experiments in air and they confirmed the A = HIL and E = SIH trends observed by Chu and Churchill, but only for Grashof numbers (5 x lo6 5 Gr, 5 9 x lo6). Keyhani et al.,5,6 studied multiple discrete heat sources mounted vertically on cavity walls, for Rayleigh numbers (based on cavity width) Rat 5 lo8. From the above studies one finds that: 1) Different length scales were used in the definitions of the Nusselt number and the Rayleigh number (see Table 1). The choice of the characteristic length, as shown in the study of Chu and Churchill, is arbitrary, but affects the results of Nusselt numbers or Rayleigh numbers significantly. 2) It is difficult to extend the numerical results to high Ra; but in this study an extrapolated correlation is given for Ra 105. r ISOTHERMAL BOUNDARY Fig. 1 Schematic of the enclosure. ADIABATIC BOUNDARY - DISCRETE HEAT SOURCE (ISOFLUX (4)) OR (ISOTHERMAL (TA)) Table 1 Scale lengths of Nu, Ra in previous studies for Ra = 0 Scale length Scale length Ref. of Nu of Ra 2 S H 3, 4 H H 5, 6 L L In the present study numerical results will be given for relative source lengths, E, of 1, 0.75, 0.5, and 0.25 over the range 0 5 Ra 5 lo6. Two limiting cases of a discrete beat source are examined: an isoflux discrete heat source (IFDHS) and an isothermal discrete heat source (ITDHS). The paper is organized as follows. In the following section, the governing equations (GE) are stated with proper assumptions. In the third section, the nondimensional form of the (GE) is discussed. The numerical solution for (GE) is presented in the fourth section. The numerical results are discussed in the fifth section in the context of the flow regimes. The obtained and correlated results are discussed in the sixth section. Finally, conclusions are given in the last section. Governing Equations and Conduction Solutions Assumptions used in the present study are: 1) Fluid is considered as incompressible and Newtonian. 2) Flow is laminar in two dimensions x and y. 3) Thermal properties of the fluid are constant except in the buoyancy term Boussinesq approximation. 4) Pressure changes inside the cavity are moderate. 5) Viscous dissipation effects are neglected. For two-dimensional, laminar natural-convection inside enclosures, the governing equations of mass, momentum, and energy, can be wtitten as follows: DU P g t = --&- apd + pv u DT pc - = kv T Dt Analytical Solution for Ra = 0 A separable series solution to Laplace s equation V2T = 0 can be used to solve the conduction problem. for Ra = 0, by taking 0 = T - T,. The form of the solution, Eq. (9, satisfies the homogeneous condition at x = 0 and the adiabatic conditions along y = 0 and H or cc O(x, Y ) = a& + a,cos(h,y)sinh(hs) (5) n=l cc (4) O(x, y) = a,+,; A, = ndh (6) n=n The temperature gradient is therefore ae - = 5 a,,+: ax The remaining boundary conditions along x = L are ae _ y5- H - S - H + S s y c H (8) ax 2 2 and for the IFDHS or ITDHS. we have ao H- S H+S 5 y 5 - ax k 2 2 (7) (9) JAN. -MARCH 1992 NATURAL CONVECTION FROM DISCRETE HEAT SOURCES 123 or H- S 5 y 5 - H+S 2 e=eh - 2 (10) Nondimensional Form of the Governing Equations The governing equations of mass, momentum, and energy Eqs. (1-4), can be written in nondimensional form as follows: The Fourier coefficients a, in Eq. (5) can be easily and accurately determined, for either the IFDHS or ITDHS, using a continuous variational approximation as noted in Lemczyk and Gladwell.' The overall enclosure resistance is defined by r, = &/Q (11) For the IFDHS, the a, in Eq. (5) are explicitly defined, since $A(L, y) is continuous-along 0 5 y 5 H. Therefore, by using the area-averaging of es(6, = 1/S J O.&y) and dividing by Q, we can write ' a ' _- Dv - -- Or ay + PrV2V + RaPrO _- Do - Or a,l 1 r,=-+- with the nondimensional variables defined as qs Sq5-E' an[sin(nn(l + ~)/2) - sin(n.rr(1 - ~)/2)]sinh(n~/A) U T - us v=- US td' 7 =- n=l 2 n a4 CYA S2 (12) x=- XA y PdPc;s2 = y'? P' = a, and a, are obtained by solving Eq. (6) at x = L and 0 S S k2a2 5 y 5 H for IFDHS T - ITDHS 9s a, = - kh Th - Tc (18) (19) a,=-c 2qH kt2 Equation (12) will be as follows: [sin(n.rr(l + ~)/2) - sin(n.rr(1 - ~)/2)] n2cosh(n da) [sin(n.rr(l +.5)/2) - sin(nr(1 - &)/2)I2tanh(nq/A) n=l 5 n3 (13) However, for the ITDHS, the a, are not explicitly defined, but can be determined using the procedure developed in Reference 7. The solution was also checked against a conformal mapping solution s and found to agree within 1% for E 5 1. In either case, however, the a,, are not explicit as in the IFDHS case. By nondimensionalizing Eq. (13), we obtain The nondimensional initial and boundary conditions are given as t=o: u = v = = O t o:x=- LA u=v=o ao S O5Y5Y1 - ax =o ao Y,5Y5Y2 -- ax- I IFDHS Y,5Y5Y2 o=1 ITDHS Y25Y5Y, -=o ax A sinz(mm)tanh(2mda) R, = $E' m=l m3 (14) where n = 2.m. From Eq. (14), one notes that the total resistance of the enclosure at Ra = 0 consists of the linear sum of the material and constriction resistances (R,,,, R,) where The local and overall thermal energy balance at x = L may be written as The scale length S/A is naturally determined from Eq. (15). It has physical meaning because it depends on the heat source size and geometry of the enclosure. Remark 1 The characteristic length SIA for an enclosure with a discrete heat source is obtained directly from the analytical solution of the governing equations when Ra = 0. The total heat flow rate at x = L, is given by!e Q = q.s.l = k.1 -dy = h.s.18, (22) H-s ax 2 124 G. REFAI AHMED AND M. M. YOVANOVICH J. THERMOPHYSICS The area-average Nusselt number is defined as Numerical Solution A numerical solution of Eqs. (16-19) is sought subject to the conditions of no slip on all solid boundaries. Used in this study is a new version of a finite difference computer code, developed by Refai' based on the Marker and Cell (MAC) method, as carried out by Hirt et a1.i The method uses a finite difference formulation with primitive variables as the dependent variables. The enclosure is, therefore, divided into a finite difference mesh of square cells (AX, AY) surrounded by a single layer of fictitious cells where the boundary conditions are imposed as shown in Fig. 2. The fluid velocity components (U, V) are defined at each cell surface while the pressure and temperature (Pd, 0) are located at the cell centers. Further details on this procedure can be found in Refs. (9) and (11). Four steps were taken to validate the MAC technique used for the numerical solutions: 1) Test the convergence. 2) Compare the velocity and the temperature profiles for different Ra with previous studies. 3) Compare results with analytical solutions at Ra = 0. 4) Compare results of Nu with previous numerical and experimental studies. Items 1 and 2 were carried out in detail by Refai' and Fath et al. Tables 2 and 3 show the comparison between the analytical and numerical results when Ra = 0. The comparison between previous and present results will be discussed later. Flow Regimes Figure 3 shows the IFDHS case of nondimensional velocity profiles of the Y-component, V, at midheight of a vertical square enclosure for different Ra*, Pr = 0.72 and E. At E = 1, as Ra* increased from lo3 (conduction regime) to lo6 (laminar regime), the position of the maximum velocity moved closer to the vertical boundaries. Although the trends throughout the full flow regime are similar, the velocity pro- Table 2 Comparison between numerical and analytical solutions (IFDHS) at Ra* = 0 E R, R, R, t% Diff. * ' * OSo * ' * o,02 l.o o 1.0 *Numerical results. *Analytical results. t% Wf. = [(R,,, - R,a,,,,)/Rt~,,,I x 100% Table 3 Comparison between numerical and analytical solutions (ITDHS) at Ra = 0 E R, R, R, % Diff. * ' * OSO * ' * o.02 l.o o *Numerical results. *Analytical results. 0 Cell No. 1 (Air) 0 Cell No. 2 (Heat Source) 0 Cell No. 3 (Cold Cell) V Cell No. 4 (Adiabatic Cell) Fig. 2 Finite difference grid with flagged boundary cells: typical cell arrangement was 1 air cell; 2 to 4 boundary cells. A = l Pr = I,I,I,I,I X X Fig. 3 Dimensionless velocity profiles at the midheight of the enclosure (IFDHS): a) E = 1, b) E = files are slightly skewed towards the cold isothermal boundary. The nondimensional temperature distributions are shown in Fig. 4. For a discrete isoflux heat source, at Ra* = lo3, a linear distribution is obtained representing the conduction regime. For Ra* = lo6 (laminar regime), the distribution shows a slight negative temperature gradient in a portion of the enclosure as a result of the fluid motion. This has also been observed in previous numerical and experimental studies -13 but for full contact, E = 1, and isothermal source. The same trend for the velocity and temperature profiles is obtained for E = 0.25 but at lower values of Ra* as shown in Figs. 3b and 4b. This indicates that the change from conduction to laminar regime for the smallest E occurs at lower Rayleigh numbers compared to E = 1. For E = 0.25 the Rayleigh numbers for conduction, transition from conduction, and laminar regimes are 3.9, 156, and 3906, respectively; I JAN. -MARCH 1992 e 0.6- Pr NATURAL CONVECTION FROM DISCRETE HEAT SOURCES I 3 Z 10 / X 1.2 Rd I Rd = A- 1 Pr = 0.72 e IO 102 lo3 lo4 io5 io6 10' RGH Fig. 5 NuRa relationships for different scale lengths. A = 1 A E = 0.75 Pr = Z 2- E = 0.25 X Fig. 4 Dimensionless temperature profiles at the Midheight of the enclosure (IFDHS) a) E = 1, b) E = however, for E = 1 they are 1000, 4 x lo4, and lo6, respectively. On the other hand, the ratio between Rayleigh numbers in conduction and transition regimes for E = 0.25 is 40, which is the same for E = 1. The same observation for the ratio between conduction and laminar regimes is found for E = 0.25, and 1; it is lo3. The same observations were found for ITDHS. Discussion of Results Effect of Scale Length The choice of scale length is very important because it alters the trend of the relationship between Nu and Ra. Figure 5a shows the relationship between Nu, and Ra: using the height H or width L of the enclosure (square enclosure) as the scale length. By this definition, when E decreases Nu, increases therefore Nu, + 30 at E On the other hand, Fig. 5b shows the relationship between Nu, and Rag for several values of the relative source size E when Nu is based on the scale length S and Ra: is based on the height of the cavity, as in Ref. (2). These relationships cannot go to one asymptote, therefore it is difficult to correlate these relationships with one equation. In contrast, Fig. 6a shows the relationships between Nu and Ra* when they are based on the scale length obtained from the analytical solution. At Ra* = 0, it can be seen that if E 1, then Nu 1, which clearly represents the conduction solution behavior. It is easily shown that the scale length used in the present study is exactly the length used in previous studies' lz-lf when the heat source completely covers the right vertical face. Remark 2 For full contact heat source, E = 1, the scale length of the present study is SIA = L which is consistent with the previous ~tudies.~j~-'~ Heat Transfer Results The results for a vertical enclosure using ITDHS or IFDHS are shown in Fig. 6a for different E. It is found for Ra* greater lol 103 lo4 io5 io6 io7 RG 1 io lo2 io3 io4 io5 io6 io7 Ra Fig. 6 Nu-Ra relationships depend on the scale length from the analytical solution. than approximately 300 that Nu decreases with increasing E, and the slope of log Nu vs log Ra* is 0.26 at E = 1 and 0.2 at E = 0.25, for IFDHS. ITDHS follows the same trend as IFDHS but the slope of Ra is 0.3 at E = 1 and 0.28 at E = Figure 6b shows good agreement between the present results and the experimental results of Eckert and Carls~n'~ at E = 1. The present results are closer to the experimental resultsi5 than the numerical results of ElderI6 also shown in Fig. 6b. The results of Chu and Churchil12 show the same trend as the present results, if the scale length is changed to the recommended scale length. Thermal Conductance and Thermal Resistance To discuss both thermal resistance and thermal conductance, a question may be posed: Is it necessary to study this problem using new concepts? It is known that Nu is proportional to Ra; when Ra -+ m, Nu will also go to infinity. Using the classical approach, it would have been difficult, if not impossible, to predict Nu beyond the range that is shown in Fig. 6. By separating the effect of conduction from the Nusselt number, the change of thermal conductance can be obtained as defined by where Nu(Ra = 0) + ANu = Nu(Ra) (24) Nu(Ra = 0) = IIR, and R, = R,, + R, Utilizing the new approach, Fig. 7 shows that, for all E, ANu increases with increasing Rayleigh. All curves of ANu 126 G. REFAI AHMED AND M. M. YOVANOVICH J. THERMOPHYSICS in.y a z' [F/:'O = 10-2 a 8 E =0.5 E = 0.25 t A = 1 I Pr = 0.72 IFDHS Correlations of NuRa Numerical Results The complex dependence of Nu on Ra and E apparently rules out obtaining a single equation that could correlate all results. Flack and Turner' obtained a separate correlation for each E but it is difficult to use them as a design equation. The correlation for multidiscrete heat sources was obtained by Keyhani et al.,s but it is limited to their study only (i.e., A = 4.5). In the present study, correlations are obtained that include the effects of the boundary conditions and the relative size of the discrete heat source. The method suggested by Churchill and Usagi17 was found to be remarkably successful in correlating rates of transfer for processes which vary uniformly between two asymptotes. A least-squares method was used to correlate the data for each asymptote as shown in Eqs. (25) and (27). In contrast, to develop Eqs. (26) and (28) for high Rayleigh numbers, we used Fig. 7 to estimate the slope of the relationship between ANu and Rayleigh. This relationship is a straight line on a log-log plot; then the effect of conduction at Ra = 0 was ad

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